Tutorial 4: 2D Kalman Filter¶
Week 3, Day 2: Hidden Dynamics
By Neuromatch Academy
Content creators: Caroline Haimerl and Byron Galbraith
Content reviewers: Jesse Livezey, Matt Krause, Michael Waskom, and Xaq Pitkow
Useful reference:
Roweis, Ghahramani (1998): A unifying review of linear Gaussian Models
Bishop (2006): Pattern Recognition and Machine Learning
Acknowledgement
This tutorial is in part based on code originally created by Caroline Haimerl for Dr. Cristina Savin’s Probabilistic Time Series class at the Center for Data Science, New York University
We will edit the above video to end at 2:20
Video 1: Introduction¶
Video available at https://youtu.be/6f_51L3i5aQ
Tutorial Objectives¶
In the previous tutorials we got an intuition for the Kalman filter in one dimension. In this tutorial, we will examine the 2D Kalman filter and more of its mathematical foundations.
In this tutorial, you will:
Review linear dynamical systems
Implement the Kalman filter
Explore how the Kalman filter can be used to smooth data from an eye-tracking experiment
import sys
!conda install -c conda-forge ipywidgets --yes
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==> WARNING: A newer version of conda exists. <==
current version: 4.9.2
latest version: 4.10.1
Please update conda by running
$ conda update -n base -c defaults conda
## Package Plan ##
environment location: /usr/share/miniconda
added / updated specs:
- ipywidgets
The following packages will be downloaded:
package | build
---------------------------|-----------------
argon2-cffi-20.1.0 | py38h25fe258_2 47 KB conda-forge
async_generator-1.10 | py_0 18 KB conda-forge
attrs-21.2.0 | pyhd8ed1ab_0 44 KB conda-forge
backcall-0.2.0 | pyh9f0ad1d_0 13 KB conda-forge
backports-1.0 | py_2 4 KB conda-forge
backports.functools_lru_cache-1.6.4| pyhd8ed1ab_0 9 KB conda-forge
bleach-3.3.0 | pyh44b312d_0 111 KB conda-forge
ca-certificates-2021.5.30 | ha878542_0 136 KB conda-forge
certifi-2021.5.30 | py38h578d9bd_0 141 KB conda-forge
conda-4.10.1 | py38h578d9bd_0 3.1 MB conda-forge
decorator-5.0.9 | pyhd8ed1ab_0 11 KB conda-forge
defusedxml-0.7.1 | pyhd8ed1ab_0 23 KB conda-forge
entrypoints-0.3 | pyhd8ed1ab_1003 8 KB conda-forge
importlib-metadata-4.5.0 | py38h578d9bd_0 31 KB conda-forge
ipykernel-5.5.5 | py38hd0cf306_0 167 KB conda-forge
ipython-7.24.1 | py38hd0cf306_0 1.1 MB conda-forge
ipython_genutils-0.2.0 | py_1 21 KB conda-forge
ipywidgets-7.6.3 | pyhd3deb0d_0 101 KB conda-forge
jedi-0.18.0 | py38h578d9bd_2 937 KB conda-forge
jinja2-2.11.3 | pyh44b312d_0 93 KB conda-forge
jsonschema-3.2.0 | pyhd8ed1ab_3 45 KB conda-forge
jupyter_client-6.1.12 | pyhd8ed1ab_0 79 KB conda-forge
jupyter_core-4.7.1 | py38h578d9bd_0 72 KB conda-forge
jupyterlab_pygments-0.1.2 | pyh9f0ad1d_0 8 KB conda-forge
jupyterlab_widgets-1.0.0 | pyhd8ed1ab_1 130 KB conda-forge
libsodium-1.0.18 | h36c2ea0_1 366 KB conda-forge
markupsafe-1.1.1 | py38h8df0ef7_2 27 KB conda-forge
matplotlib-inline-0.1.2 | pyhd8ed1ab_2 11 KB conda-forge
mistune-0.8.4 |py38h25fe258_1002 54 KB conda-forge
nbclient-0.5.3 | pyhd8ed1ab_0 67 KB conda-forge
nbconvert-6.1.0 | py38h578d9bd_0 521 KB conda-forge
nbformat-5.1.3 | pyhd8ed1ab_0 47 KB conda-forge
nest-asyncio-1.5.1 | pyhd8ed1ab_0 9 KB conda-forge
notebook-6.4.0 | pyha770c72_0 6.1 MB conda-forge
openssl-1.1.1k | h27cfd23_0 2.5 MB
packaging-20.9 | pyh44b312d_0 35 KB conda-forge
pandoc-2.14.0.3 | h7f98852_0 12.0 MB conda-forge
pandocfilters-1.4.2 | py_1 9 KB conda-forge
parso-0.8.2 | pyhd8ed1ab_0 68 KB conda-forge
pexpect-4.8.0 | pyh9f0ad1d_2 47 KB conda-forge
pickleshare-0.7.5 | py_1003 9 KB conda-forge
prometheus_client-0.11.0 | pyhd8ed1ab_0 46 KB conda-forge
prompt-toolkit-3.0.19 | pyha770c72_0 244 KB conda-forge
ptyprocess-0.7.0 | pyhd3deb0d_0 16 KB conda-forge
pygments-2.9.0 | pyhd8ed1ab_0 754 KB conda-forge
pyparsing-2.4.7 | pyh9f0ad1d_0 60 KB conda-forge
pyrsistent-0.17.3 | py38h25fe258_1 89 KB conda-forge
python-dateutil-2.8.1 | py_0 220 KB conda-forge
python_abi-3.8 | 2_cp38 4 KB conda-forge
pyzmq-19.0.2 | py38ha71036d_2 511 KB conda-forge
send2trash-1.7.1 | pyhd8ed1ab_0 17 KB conda-forge
terminado-0.10.1 | py38h578d9bd_0 26 KB conda-forge
testpath-0.5.0 | pyhd8ed1ab_0 86 KB conda-forge
tornado-6.1 | py38h25fe258_0 646 KB conda-forge
traitlets-5.0.5 | py_0 81 KB conda-forge
wcwidth-0.2.5 | pyh9f0ad1d_2 33 KB conda-forge
webencodings-0.5.1 | py_1 12 KB conda-forge
widgetsnbextension-3.5.1 | py38h578d9bd_4 1.8 MB conda-forge
zeromq-4.3.4 | h2531618_0 331 KB
zipp-3.4.1 | pyhd8ed1ab_0 11 KB conda-forge
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Total: 33.2 MB
The following NEW packages will be INSTALLED:
argon2-cffi conda-forge/linux-64::argon2-cffi-20.1.0-py38h25fe258_2
async_generator conda-forge/noarch::async_generator-1.10-py_0
attrs conda-forge/noarch::attrs-21.2.0-pyhd8ed1ab_0
backcall conda-forge/noarch::backcall-0.2.0-pyh9f0ad1d_0
backports conda-forge/noarch::backports-1.0-py_2
backports.functoo~ conda-forge/noarch::backports.functools_lru_cache-1.6.4-pyhd8ed1ab_0
bleach conda-forge/noarch::bleach-3.3.0-pyh44b312d_0
decorator conda-forge/noarch::decorator-5.0.9-pyhd8ed1ab_0
defusedxml conda-forge/noarch::defusedxml-0.7.1-pyhd8ed1ab_0
entrypoints conda-forge/noarch::entrypoints-0.3-pyhd8ed1ab_1003
importlib-metadata conda-forge/linux-64::importlib-metadata-4.5.0-py38h578d9bd_0
ipykernel conda-forge/linux-64::ipykernel-5.5.5-py38hd0cf306_0
ipython conda-forge/linux-64::ipython-7.24.1-py38hd0cf306_0
ipython_genutils conda-forge/noarch::ipython_genutils-0.2.0-py_1
ipywidgets conda-forge/noarch::ipywidgets-7.6.3-pyhd3deb0d_0
jedi conda-forge/linux-64::jedi-0.18.0-py38h578d9bd_2
jinja2 conda-forge/noarch::jinja2-2.11.3-pyh44b312d_0
jsonschema conda-forge/noarch::jsonschema-3.2.0-pyhd8ed1ab_3
jupyter_client conda-forge/noarch::jupyter_client-6.1.12-pyhd8ed1ab_0
jupyter_core conda-forge/linux-64::jupyter_core-4.7.1-py38h578d9bd_0
jupyterlab_pygmen~ conda-forge/noarch::jupyterlab_pygments-0.1.2-pyh9f0ad1d_0
jupyterlab_widgets conda-forge/noarch::jupyterlab_widgets-1.0.0-pyhd8ed1ab_1
libsodium conda-forge/linux-64::libsodium-1.0.18-h36c2ea0_1
markupsafe conda-forge/linux-64::markupsafe-1.1.1-py38h8df0ef7_2
matplotlib-inline conda-forge/noarch::matplotlib-inline-0.1.2-pyhd8ed1ab_2
mistune conda-forge/linux-64::mistune-0.8.4-py38h25fe258_1002
nbclient conda-forge/noarch::nbclient-0.5.3-pyhd8ed1ab_0
nbconvert conda-forge/linux-64::nbconvert-6.1.0-py38h578d9bd_0
nbformat conda-forge/noarch::nbformat-5.1.3-pyhd8ed1ab_0
nest-asyncio conda-forge/noarch::nest-asyncio-1.5.1-pyhd8ed1ab_0
notebook conda-forge/noarch::notebook-6.4.0-pyha770c72_0
packaging conda-forge/noarch::packaging-20.9-pyh44b312d_0
pandoc conda-forge/linux-64::pandoc-2.14.0.3-h7f98852_0
pandocfilters conda-forge/noarch::pandocfilters-1.4.2-py_1
parso conda-forge/noarch::parso-0.8.2-pyhd8ed1ab_0
pexpect conda-forge/noarch::pexpect-4.8.0-pyh9f0ad1d_2
pickleshare conda-forge/noarch::pickleshare-0.7.5-py_1003
prometheus_client conda-forge/noarch::prometheus_client-0.11.0-pyhd8ed1ab_0
prompt-toolkit conda-forge/noarch::prompt-toolkit-3.0.19-pyha770c72_0
ptyprocess conda-forge/noarch::ptyprocess-0.7.0-pyhd3deb0d_0
pygments conda-forge/noarch::pygments-2.9.0-pyhd8ed1ab_0
pyparsing conda-forge/noarch::pyparsing-2.4.7-pyh9f0ad1d_0
pyrsistent conda-forge/linux-64::pyrsistent-0.17.3-py38h25fe258_1
python-dateutil conda-forge/noarch::python-dateutil-2.8.1-py_0
python_abi conda-forge/linux-64::python_abi-3.8-2_cp38
pyzmq conda-forge/linux-64::pyzmq-19.0.2-py38ha71036d_2
send2trash conda-forge/noarch::send2trash-1.7.1-pyhd8ed1ab_0
terminado conda-forge/linux-64::terminado-0.10.1-py38h578d9bd_0
testpath conda-forge/noarch::testpath-0.5.0-pyhd8ed1ab_0
tornado conda-forge/linux-64::tornado-6.1-py38h25fe258_0
traitlets conda-forge/noarch::traitlets-5.0.5-py_0
wcwidth conda-forge/noarch::wcwidth-0.2.5-pyh9f0ad1d_2
webencodings conda-forge/noarch::webencodings-0.5.1-py_1
widgetsnbextension conda-forge/linux-64::widgetsnbextension-3.5.1-py38h578d9bd_4
zeromq pkgs/main/linux-64::zeromq-4.3.4-h2531618_0
zipp conda-forge/noarch::zipp-3.4.1-pyhd8ed1ab_0
The following packages will be UPDATED:
ca-certificates pkgs/main::ca-certificates-2020.10.14~ --> conda-forge::ca-certificates-2021.5.30-ha878542_0
certifi pkgs/main/noarch::certifi-2020.6.20-p~ --> conda-forge/linux-64::certifi-2021.5.30-py38h578d9bd_0
conda pkgs/main::conda-4.9.2-py38h06a4308_0 --> conda-forge::conda-4.10.1-py38h578d9bd_0
openssl 1.1.1h-h7b6447c_0 --> 1.1.1k-h27cfd23_0
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!conda install numpy matplotlib scipy requests --yes
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## Package Plan ##
environment location: /usr/share/miniconda
added / updated specs:
- matplotlib
- numpy
- requests
- scipy
The following packages will be downloaded:
package | build
---------------------------|-----------------
blas-1.0 | mkl 6 KB
ca-certificates-2021.5.25 | h06a4308_1 112 KB
certifi-2021.5.30 | py38h06a4308_0 138 KB
conda-4.10.1 | py38h06a4308_1 2.9 MB
cycler-0.10.0 | py38_0 14 KB
dbus-1.13.18 | hb2f20db_0 504 KB
expat-2.4.1 | h2531618_2 168 KB
fontconfig-2.13.1 | h6c09931_0 250 KB
freetype-2.10.4 | h5ab3b9f_0 596 KB
glib-2.68.2 | h36276a3_0 3.0 MB
gst-plugins-base-1.14.0 | h8213a91_2 4.9 MB
gstreamer-1.14.0 | h28cd5cc_2 3.2 MB
icu-58.2 | he6710b0_3 10.5 MB
intel-openmp-2021.2.0 | h06a4308_610 1.3 MB
jpeg-9b | h024ee3a_2 214 KB
kiwisolver-1.3.1 | py38h2531618_0 80 KB
lcms2-2.12 | h3be6417_0 312 KB
libgfortran-ng-7.5.0 | ha8ba4b0_17 22 KB
libgfortran4-7.5.0 | ha8ba4b0_17 995 KB
libpng-1.6.37 | hbc83047_0 278 KB
libtiff-4.2.0 | h85742a9_0 502 KB
libuuid-1.0.3 | h1bed415_2 15 KB
libwebp-base-1.2.0 | h27cfd23_0 437 KB
libxcb-1.14 | h7b6447c_0 505 KB
libxml2-2.9.10 | hb55368b_3 1.2 MB
lz4-c-1.9.3 | h2531618_0 186 KB
matplotlib-3.3.4 | py38h06a4308_0 26 KB
matplotlib-base-3.3.4 | py38h62a2d02_0 5.1 MB
mkl-2021.2.0 | h06a4308_296 144.3 MB
mkl-service-2.3.0 | py38h27cfd23_1 57 KB
mkl_fft-1.3.0 | py38h42c9631_2 180 KB
mkl_random-1.2.1 | py38ha9443f7_2 305 KB
numpy-1.20.2 | py38h2d18471_0 23 KB
numpy-base-1.20.2 | py38hfae3a4d_0 4.6 MB
olefile-0.46 | py_0 33 KB
pcre-8.45 | h295c915_0 207 KB
pillow-8.2.0 | py38he98fc37_0 628 KB
pyqt-5.9.2 | py38h05f1152_4 4.5 MB
qt-5.9.7 | h5867ecd_1 68.5 MB
requests-2.25.1 | pyhd3eb1b0_0 52 KB
scipy-1.6.2 | py38had2a1c9_1 15.6 MB
sip-4.19.13 | py38he6710b0_0 277 KB
zstd-1.4.9 | haebb681_0 480 KB
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Total: 277.0 MB
The following NEW packages will be INSTALLED:
blas pkgs/main/linux-64::blas-1.0-mkl
cycler pkgs/main/linux-64::cycler-0.10.0-py38_0
dbus pkgs/main/linux-64::dbus-1.13.18-hb2f20db_0
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# Install PyKalman (https://pykalman.github.io/)
!pip install pykalman --quiet
# Imports
import numpy as np
import matplotlib.pyplot as plt
import pykalman
from scipy import stats
Figure settings¶
#@title Figure settings
import ipywidgets as widgets # interactive display
%config InlineBackend.figure_format = 'retina'
plt.style.use("https://raw.githubusercontent.com/NeuromatchAcademy/course-content/master/nma.mplstyle")
Data retrieval and loading¶
#@title Data retrieval and loading
import io
import os
import hashlib
import requests
fname = "W2D3_mit_eyetracking_2009.npz"
url = "https://osf.io/jfk8w/download"
expected_md5 = "20c7bc4a6f61f49450997e381cf5e0dd"
if not os.path.isfile(fname):
try:
r = requests.get(url)
except requests.ConnectionError:
print("!!! Failed to download data !!!")
else:
if r.status_code != requests.codes.ok:
print("!!! Failed to download data !!!")
elif hashlib.md5(r.content).hexdigest() != expected_md5:
print("!!! Data download appears corrupted !!!")
else:
with open(fname, "wb") as fid:
fid.write(r.content)
def load_eyetracking_data(data_fname=fname):
with np.load(data_fname, allow_pickle=True) as dobj:
data = dict(**dobj)
images = [plt.imread(io.BytesIO(stim), format='JPG')
for stim in data['stimuli']]
subjects = data['subjects']
return subjects, images
Helper functions¶
#@title Helper functions
np.set_printoptions(precision=3)
def plot_kalman(state, observation, estimate=None, label='filter', color='r-',
title='LDS', axes=None):
if axes is None:
fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(16, 6))
ax1.plot(state[:, 0], state[:, 1], 'g-', label='true latent')
ax1.plot(observation[:, 0], observation[:, 1], 'k.', label='data')
else:
ax1, ax2 = axes
if estimate is not None:
ax1.plot(estimate[:, 0], estimate[:, 1], color=color, label=label)
ax1.set(title=title, xlabel='X position', ylabel='Y position')
ax1.legend()
if estimate is None:
ax2.plot(state[:, 0], observation[:, 0], '.k', label='dim 1')
ax2.plot(state[:, 1], observation[:, 1], '.', color='grey', label='dim 2')
ax2.set(title='correlation', xlabel='latent', ylabel='measured')
else:
ax2.plot(state[:, 0], estimate[:, 0], '.', color=color,
label='latent dim 1')
ax2.plot(state[:, 1], estimate[:, 1], 'x', color=color,
label='latent dim 2')
ax2.set(title='correlation',
xlabel='real latent',
ylabel='estimated latent')
ax2.legend()
return ax1, ax2
def plot_gaze_data(data, img=None, ax=None):
# overlay gaze on stimulus
if ax is None:
fig, ax = plt.subplots(figsize=(8, 6))
xlim = None
ylim = None
if img is not None:
ax.imshow(img, aspect='auto')
ylim = (img.shape[0], 0)
xlim = (0, img.shape[1])
ax.scatter(data[:, 0], data[:, 1], c='m', s=100, alpha=0.7)
ax.set(xlim=xlim, ylim=ylim)
return ax
def plot_kf_state(kf, data, ax):
mu_0 = np.ones(kf.n_dim_state)
mu_0[:data.shape[1]] = data[0]
kf.initial_state_mean = mu_0
mu, sigma = kf.smooth(data)
ax.plot(mu[:, 0], mu[:, 1], 'limegreen', linewidth=3, zorder=1)
ax.scatter(mu[0, 0], mu[0, 1], c='orange', marker='>', s=200, zorder=2)
ax.scatter(mu[-1, 0], mu[-1, 1], c='orange', marker='s', s=200, zorder=2)
Section 1: Linear Dynamical System (LDS)¶
The below video will be edited to 0:33 - 1:09, and then 2:01 - 3:32
Video 2: Linear Dynamical Systems¶
Video available at https://youtu.be/2SWh639YgEg
Kalman filter definitions:¶
The latent state \(s_t\) evolves as a stochastic linear dynamical system in discrete time, with a dynamics matrix \(D\): $\(s_t = Ds_{t-1}+w_t\)$
Just as in the HMM, the structure is a Markov chain where the state at time point \(t\) is conditionally independent of previous states given the state at time point \(t-1\).
Sensory measurements \(m_t\) (observations) are noisy linear projections of the latent state: $\(m_t = Hs_{t}+\eta_t\)$
Both states and measurements have Gaussian variability, often called noise: ‘process noise’ \(w_t\) for the states, and ‘measurement’ or ‘observation noise’ \(\eta_t\) for the measurements. The initial state is also Gaussian distributed. These quantites have means and covariances:
As a consequence, \(s_t\), \(m_t\) and their joint distributions are Gaussian. This makes all of the math analytically tractable using linear algebra, so we can easily compute the marginal and conditional distributions we will use for inferring the current state given the entire history of measurements.
Please note: we are trying to create uniform notation across tutorials. In some videos created in 2020, measurements \(m_t\) were denoted \(y_t\), and the Dynamics matrix \(D\) was denoted \(F\). We apologize for any confusion!
Section 1.1: Sampling from a latent linear dynamical system¶
The first thing we will investigate is how to generate timecourse samples from a linear dynamical system given its parameters. We will start by defining the following system:
# task dimensions
n_dim_state = 2
n_dim_obs = 2
# initialize model parameters
params = {
'D': 0.9 * np.eye(n_dim_state), # state transition matrix
'Q': np.eye(n_dim_obs), # state noise covariance
'H': np.eye(n_dim_state), # observation matrix
'R': 1.0 * np.eye(n_dim_obs), # observation noise covariance
'mu_0': np.zeros(n_dim_state), # initial state mean
'sigma_0': 0.1 * np.eye(n_dim_state), # initial state noise covariance
}
Coding note: We used a parameter dictionary params above. As the number of parameters we need to provide to our functions increases, it can be beneficial to condense them into a data structure like this to clean up the number of inputs we pass in. The trade-off is that we have to know what is in our data structure to use those values, rather than looking at the function signature directly.
Exercise 1: Sampling from a linear dynamical system¶
In this exercise you will implement the dynamics functions of a linear dynamical system to sample both a latent space trajectory (given parameters set above) and noisy measurements.
def sample_lds(n_timesteps, params, seed=0):
""" Generate samples from a Linear Dynamical System specified by the provided
parameters.
Args:
n_timesteps (int): the number of time steps to simulate
params (dict): a dictionary of model paramters: (D, Q, H, R, mu_0, sigma_0)
seed (int): a random seed to use for reproducibility checks
Returns:
ndarray, ndarray: the generated state and observation data
"""
n_dim_state = params['D'].shape[0]
n_dim_obs = params['H'].shape[0]
# set seed
np.random.seed(seed)
# precompute random samples from the provided covariance matrices
# mean defaults to 0
mi = stats.multivariate_normal(cov=params['Q']).rvs(n_timesteps)
eta = stats.multivariate_normal(cov=params['R']).rvs(n_timesteps)
# initialize state and observation arrays
state = np.zeros((n_timesteps, n_dim_state))
obs = np.zeros((n_timesteps, n_dim_obs))
###################################################################
## TODO for students: compute the next state and observation values
# Fill out function and remove
raise NotImplementedError("Student excercise: compute the next state and observation values")
###################################################################
# simulate the system
for t in range(n_timesteps):
# write the expressions for computing state values given the time step
if t == 0:
state[t] = ...
else:
state[t] = ...
# write the expression for computing the observation
obs[t] = ...
return state, obs
# Uncomment below to test your function
# state, obs = sample_lds(100, params)
# print('sample at t=3 ', state[3])
# plot_kalman(state, obs, title='sample')
Example output:
Interactive Demo: Adjusting System Dynamics¶
To test your understanding of the parameters of a linear dynamical system, think about what you would expect if you made the following changes:
Reduce observation noise \(R\)
Increase respective temporal dynamics \(D\)
Use the interactive widget below to vary the values of \(R\) and \(D\).
¶
#@title
#@markdown Make sure you execute this cell to enable the widget!
@widgets.interact(R=widgets.FloatLogSlider(1., min=-2, max=2),
D=widgets.FloatSlider(0.9, min=0.0, max=1.0, step=.01))
def explore_dynamics(R=0.1, D=0.5):
params = {
'D': D * np.eye(n_dim_state), # state transition matrix
'Q': np.eye(n_dim_obs), # state noise covariance
'H': np.eye(n_dim_state), # observation matrix
'R': R * np.eye(n_dim_obs), # observation noise covariance
'mu_0': np.zeros(n_dim_state), # initial state mean,
'sigma_0': 0.1 * np.eye(n_dim_state), # initial state noise covariance
}
state, obs = sample_lds(100, params)
plot_kalman(state, obs, title='sample')
Section 2: Kalman Filtering¶
Video 3: Kalman Filtering¶
Video available at https://youtu.be/VboZOV9QMOI
VIDEO TO BE RE-RECORDED to improve notation and explanation, using the Information form of the Gaussians for combining likelihood with prior
We want to infer the latent state variable \(s_t\) given the measured (observed) variable \(m_t\).
First we obtain estimates of the latent state by running the filtering from \(t=0,....T\).
Where \(\hat{\mu}_t^{\rm pred}\) and \(\hat{\Sigma}_t^{\rm pred}\) are derived as follows:
This is the prediction for \(s_t\) obtained simply by taking the expected value of \(s_{t-1}\) and projecting it forward one step using the transition matrix \(D\). We do the same for the covariance, taking into account the noise covariance \(Q\) and the fact that scaling a variable by \(D\) scales its covariance \(\Sigma\) as \(D\Sigma D^T\):
We then use a Bayesian update from the newest measurements to obtain \(\hat{\mu}_t^{\rm filter}\) and \(\hat{\Sigma}_t^{\rm filter}\)
Project our prediction to observational space: $\(m_t^{\rm pred}\sim \mathcal{N}(H\hat{\mu}_t^{\rm pred}, H\hat{\Sigma}_t^{\rm pred}H^T+R)\)$
update prediction by actual data:
Kalman gain matrix: $\(K_t=\hat{\Sigma}_t^{\rm pred}H^T(H\hat{\Sigma}_t^{\rm pred}H^T+R)^{-1}\)$
We use the latent-only prediction to project it to the observational space and compute a correction proportional to the error \(m_t-HDz_{t-1}\) between prediction and data. The coefficient of this correction is the Kalman gain matrix.
Interpretations
If measurement noise is small and dynamics are fast, then estimation will depend mostly on currently observed data. If the measurement noise is large, then the Kalman filter uses past observations as well, combining them as long as the underlying state is at least somewhat predictable.
In order to explore the impact of filtering, we will use the following noisy oscillatory system:
# task dimensions
n_dim_state = 2
n_dim_obs = 2
T=100
# initialize model parameters
params = {
'D': np.array([[1., 1.], [-(2*np.pi/20.)**2., .9]]), # state transition matrix
'Q': np.eye(n_dim_obs), # state noise covariance
'H': np.eye(n_dim_state), # observation matrix
'R': 100.0 * np.eye(n_dim_obs), # observation noise covariance
'mu_0': np.zeros(n_dim_state), # initial state mean
'sigma_0': 0.1 * np.eye(n_dim_state), # initial state noise covariance
}
state, obs = sample_lds(T, params)
plot_kalman(state, obs, title='sample')
Exercise 2: Implement Kalman filtering¶
In this exercise you will implement the Kalman filter (forward) process. Your focus will be on writing the expressions for the Kalman gain, filter mean, and filter covariance at each time step (refer to the equations above).
def kalman_filter(data, params):
""" Perform Kalman filtering (forward pass) on the data given the provided
system parameters.
Args:
data (ndarray): a sequence of osbervations of shape(n_timesteps, n_dim_obs)
params (dict): a dictionary of model paramters: (D, Q, H, R, mu_0, sigma_0)
Returns:
ndarray, ndarray: the filtered system means and noise covariance values
"""
# pulled out of the params dict for convenience
D = params['D']
Q = params['Q']
H = params['H']
R = params['R']
n_dim_state = D.shape[0]
n_dim_obs = H.shape[0]
I = np.eye(n_dim_state) # identity matrix
# state tracking arrays
mu = np.zeros((len(data), n_dim_state))
sigma = np.zeros((len(data), n_dim_state, n_dim_state))
# filter the data
for t, y in enumerate(data):
if t == 0:
mu_pred = params['mu_0']
sigma_pred = params['sigma_0']
else:
mu_pred = D @ mu[t-1]
sigma_pred = D @ sigma[t-1] @ D.T + Q
###########################################################################
## TODO for students: compute the filtered state mean and covariance values
# Fill out function and remove
raise NotImplementedError("Student excercise: compute the filtered state mean and covariance values")
###########################################################################
# write the expression for computing the Kalman gain
K = ...
# write the expression for computing the filtered state mean
mu[t] = ...
# write the expression for computing the filtered state noise covariance
sigma[t] = ...
return mu, sigma
# Uncomment below to test your function
# filtered_state_means, filtered_state_covariances = kalman_filter(obs, params)
# plot_kalman(state, obs, filtered_state_means, title="my kf-filter",
# color='r', label='my kf-filter')
Example output:
Section 3: Fitting Eye Gaze Data¶
Video 4: Fitting Eye Gaze Data¶
Video available at https://youtu.be/M7OuXmVWHGI
Tracking eye gaze is used in both experimental and user interface applications. Getting an accurate estimation of where someone is looking on a screen in pixel coordinates can be challenging, however, due to the various sources of noise inherent in obtaining these measurements. A main source of noise is the general accuracy of the eye tracker device itself and how well it maintains calibration over time. Changes in ambient light or subject position can further reduce accuracy of the sensor. Eye blinks introduce a different form of noise as interruptions in the data stream which also need to be addressed.
Fortunately we have a candidate solution for handling noisy eye gaze data in the Kalman filter we just learned about. Let’s look at how we can apply these methods to a small subset of data taken from the MIT Eyetracking Database [Judd et al. 2009]. This data was collected as part of an effort to model visual saliency – given an image, can we predict where a person is most likely going to look.
# load eyetracking data
subjects, images = load_eyetracking_data()
Interactive Demo: Tracking Eye Gaze¶
We have three stimulus images and five different subjects’ gaze data. Each subject fixated in the center of the screen before the image appeared, then had a few seconds to freely look around. You can use the widget below to see how different subjects visually scanned the presented image. A subject ID of -1 will show the stimulus images without any overlayed gaze trace.
Note that the images are rescaled below for display purposes, they were in their original aspect ratio during the task itself.
¶
#@title
#@markdown Make sure you execute this cell to enable the widget!
@widgets.interact(subject_id=widgets.IntSlider(-1, min=-1, max=4),
image_id=widgets.IntSlider(0, min=0, max=2))
def plot_subject_trace(subject_id=-1, image_id=0):
if subject_id == -1:
subject = np.zeros((3, 0, 2))
else:
subject = subjects[subject_id]
data = subject[image_id]
img = images[image_id]
fig, ax = plt.subplots()
ax.imshow(img, aspect='auto')
ax.scatter(data[:, 0], data[:, 1], c='m', s=100, alpha=0.7)
ax.set(xlim=(0, img.shape[1]), ylim=(img.shape[0], 0))
Section 3.1: Fitting data with pykalman¶
Now that we have data, we’d like to use Kalman filtering to give us a better estimate of the true gaze. Up until this point we’ve known the parameters of our LDS, but here we need to estimate them from data directly. We will use the pykalman package to handle this estimation using the EM algorithm, a useful and influential learning algorithm described briefly in the bonus material.
Before exploring fitting models with pykalman it’s worth pointing out some naming conventions used by the library:
The first thing we need to do is provide a guess at the dimensionality of the latent state. Let’s start by assuming the dynamics line-up directly with the observation data (pixel x,y-coordinates), and so we have a state dimension of 2.
We also need to decide which parameters we want the EM algorithm to fit. In this case, we will let the EM algorithm discover the dynamics parameters i.e. the \(D\), \(Q\), \(H\), and \(R\) matrices.
We set up our pykalman KalmanFilter object with these settings using the code below.
# set up our KalmanFilter object and tell it which parameters we want to
# estimate
np.random.seed(1)
n_dim_obs = 2
n_dim_state = 2
kf = pykalman.KalmanFilter(
n_dim_state=n_dim_state,
n_dim_obs=n_dim_obs,
em_vars=['transition_matrices', 'transition_covariance',
'observation_matrices', 'observation_covariance']
)
Because we know from the reported experimental design that subjects fixated in the center of the screen right before the image appears, we can set the initial starting state estimate \(\mu_0\) as being the center pixel of the stimulus image (the first data point in this sample dataset) with a correspondingly low initial noise covariance \(\Sigma_0\). Once we have everything set, it’s time to fit some data.
# Choose a subject and stimulus image
subject_id = 1
image_id = 2
data = subjects[subject_id][image_id]
# Provide the initial states
kf.initial_state_mean = data[0]
kf.initial_state_covariance = 0.1*np.eye(n_dim_state)
# Estimate the parameters from data using the EM algorithm
kf.em(data)
print(f'D=\n{kf.transition_matrices}')
print(f'Q =\n{kf.transition_covariance}')
print(f'H =\n{kf.observation_matrices}')
print(f'R =\n{kf.observation_covariance}')
D=
[[ 1.004 -0.01 ]
[ 0.005 0.989]]
Q =
[[278.016 219.292]
[219.292 389.774]]
H =
[[ 0.999 0.003]
[-0.004 1.01 ]]
R =
[[26.026 19.596]
[19.596 26.745]]
We see that the EM algorithm has found fits for the various dynamics parameters. One thing you will note is that both the state and observation matrices are close to the identity matrix, which means the x- and y-coordinate dynamics are independent of each other and primarily impacted by the noise covariances.
We can now use this model to smooth the observed data from the subject. In addition to the source image, we can also see how this model will work with the gaze recorded by the same subject on the other images as well, or even with different subjects.
Below are the three stimulus images overlayed with recorded gaze in magenta and smoothed state from the filter in green, with gaze begin (orange triangle) and gaze end (orange square) markers.
¶
#@title
#@markdown Make sure you execute this cell to enable the widget!
@widgets.interact(subject_id=widgets.IntSlider(1, min=0, max=4))
def plot_smoothed_traces(subject_id=0):
subject = subjects[subject_id]
fig, axes = plt.subplots(ncols=3, figsize=(18, 4))
for data, img, ax in zip(subject, images, axes):
ax = plot_gaze_data(data, img=img, ax=ax)
plot_kf_state(kf, data, ax)
Discussion questions:¶
Why do you think one trace from one subject was sufficient to provide a decent fit across all subjects? If you were to go back and change the subject_id and/or image_id for when we fit the data using EM, do you think the fits would be different?
We don’t think the eye is exactly following a linear dynamical system. Nonetheless that is what we assumed for this exercise when we applied a Kalman filter. Despite the mismatch, these algorithms do perform well. Discuss what differences we might find between the true and assumed processes. What mistakes might be likely consequences of these differences?
Finally, recall that the original task was to use this data to help devlop models of visual salience. While our Kalman filter is able to provide smooth estimates of observed gaze data, it’s not telling us anything about why the gaze is going in a certain direction. In fact, if we sample data from our parameters and plot them, we get what amounts to a random walk.
kf_state, kf_data = kf.sample(len(data))
ax = plot_gaze_data(kf_data, img=images[2])
plot_kf_state(kf, kf_data, ax)
This should not be surprising, as we have given the model no other observed data beyond the pixels at which gaze was detected. We expect there is some other aspect driving the latent state of where to look next other than just the previous fixation location.
In summary, while the Kalman filter is a good option for smoothing the gaze trajectory itself, especially if using a lower-quality eye tracker or in noisy environmental conditions, a linear dynamical system may not be the right way to approach the much more challenging task of modeling visual saliency.
Handling Eye Blinks¶
In the MIT Eyetracking Database, raw tracking data includes times when the subject blinked. The way this is represented in the data stream is via negative pixel coordinate values.
We could try to mitigate these samples by simply deleting them from the stream, though this introduces other issues. For instance, if each sample corresponds to a fixed time step, and you arbitrarily remove some samples, the integrity of that consistent timestep between samples is lost. It’s sometimes better to flag data as missing rather than to pretend it was never there at all, especially with time series data.
Another solution is to used masked arrays. In numpy, a masked array is an ndarray with an additional embedded boolean masking array that indicates which elements should be masked. When computation is performed on the array, the masked elements are ignored. Both matplotlib and pykalman work with masked arrays, and, in fact, this is the approach taken with the data we explore in this notebook.
In preparing the dataset for this noteook, the original dataset was preprocessed to set all gaze data as masked arrays, with the mask enabled for any pixel with a negative x or y coordinate.
Bonus¶
Review on Gaussian joint, marginal and conditional distributions¶
Assume
then the marginal distributions are
and the conditional distributions are
important take away: given the joint Gaussian distribution we can derive the conditionals
Kalman Smoothing¶
Video 5: Kalman Smoothing and the EM Algorithm¶
Video available at https://youtu.be/4Ar2mYz1Nms
Obtain estimates by propagating from \(y_T\) back to \(y_0\) using results of forward pass (\(\hat{\mu}_t^{\rm filter}, \hat{\Sigma}_t^{\rm filter}, P_t=\hat{\Sigma}_{t+1}^{\rm pred}\))
This gives us the final estimate for \(z_t\).
Exercise 3: Implement Kalman smoothing¶
In this exercise you will implement the Kalman smoothing (backward) process. Again you will focus on writing the expressions for computing the smoothed mean, smoothed covariance, and \(J_t\) values.
def kalman_smooth(data, params):
""" Perform Kalman smoothing (backward pass) on the data given the provided
system parameters.
Args:
data (ndarray): a sequence of osbervations of shape(n_timesteps, n_dim_obs)
params (dict): a dictionary of model paramters: (D, Q, H, R, mu_0, sigma_0)
Returns:
ndarray, ndarray: the smoothed system means and noise covariance values
"""
# pulled out of the params dict for convenience
D= params['D']
Q = params['Q']
H = params['H']
R = params['R']
n_dim_state = D.shape[0]
n_dim_obs = H.shape[0]
# first run the forward pass to get the filtered means and covariances
mu, sigma = kalman_filter(data, params)
# initialize state mean and covariance estimates
mu_hat = np.zeros_like(mu)
sigma_hat = np.zeros_like(sigma)
mu_hat[-1] = mu[-1]
sigma_hat[-1] = sigma[-1]
# smooth the data
for t in reversed(range(len(data)-1)):
sigma_pred = D@ sigma[t] @ D.T + Q # sigma_pred at t+1
###########################################################################
## TODO for students: compute the smoothed state mean and covariance values
# Fill out function and remove
raise NotImplementedError("Student excercise: compute the smoothed state mean and covariance values")
###########################################################################
# write the expression to compute the Kalman gain for the backward process
J = ...
# write the expression to compute the smoothed state mean estimate
mu_hat[t] = ...
# write the expression to compute the smoothed state noise covariance estimate
sigma_hat[t] = ...
return mu_hat, sigma_hat
# Uncomment once the kalman_smooth function is complete
# smoothed_state_means, smoothed_state_covariances = kalman_smooth(obs, params)
# axes = plot_kalman(state, obs, filtered_state_means, color="r",
# label="my kf-filter")
# plot_kalman(state, obs, smoothed_state_means, color="b",
# label="my kf-smoothed", axes=axes)
Example output:
Forward vs Backward
Now that we have implementations for both, let’s compare their peformance by computing the MSE between the filtered (forward) and smoothed (backward) estimated states and the true latent state.
print(f"Filtered MSE: {np.mean((state - filtered_state_means)**2):.3f}")
print(f"Smoothed MSE: {np.mean((state - smoothed_state_means)**2):.3f}")
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-25-63c53cc818d2> in <module>
----> 1 print(f"Filtered MSE: {np.mean((state - filtered_state_means)**2):.3f}")
2 print(f"Smoothed MSE: {np.mean((state - smoothed_state_means)**2):.3f}")
NameError: name 'state' is not defined
In this example, the smoothed estimate is clearly superior to the filtered one. This makes sense as the forward pass uses only the past measurements, whereas the backward pass can use future measurement too, correcting the forward pass estimates given all the data we’ve collected.
So why would you ever use Kalman filtering alone, without smoothing? As Kalman filtering only depends on already observed data (i.e. the past) it can be run in a streaming, or on-line, setting. Kalman smoothing relies on future data as it were, and as such can only be applied in a batch, or off-line, setting. So use Kalman filtering if you need real-time corrections and Kalman smoothing if you are considering already-collected data.
This online case is typically what the brain faces.
The Expectation-Maximization (EM) Algorithm¶
want to maximize \(\log p(m|\theta)\)
need to marginalize out latent state (which is not tractable)
add a probability distribution \(q(s)\) which will approximate the latent state distribution
can be rewritten as
\(\mathcal{L}(q,\theta)\) contains the joint distribution of \(m\) and \(s\)
\(KL(q||p)\) contains the conditional distribution of \(s|m\)
Expectation step¶
parameters are kept fixed
find a good approximation \(q(s)\): maximize lower bound \(\mathcal{L}(q,\theta)\) with respect to \(q(s)\)
(already implemented Kalman filter+smoother)
Maximization step¶
keep distribution \(q(s)\) fixed
change parameters to maximize the lower bound \(\mathcal{L}(q,\theta)\)
As mentioned, we have already effectively solved for the E-Step with our Kalman filter and smoother. The M-step requires further derivation, which is covered in the Appendix. Rather than having you implement the M-Step yourselves, let’s instead turn to using a library that has already implemented EM for exploring some experimental data from cognitive neuroscience.
The M-step for a LDS¶
(see Bishop, chapter 13.3.2 Learning in LDS) Update parameters of the probability distribution
For the updates in the M-step we will need the following posterior marginals obtained from the Kalman smoothing results \(\hat{\mu}_t^{\rm smooth}, \hat{\Sigma}_t^{\rm smooth}\)
Update parameters
Initial parameters $$